Finite Group Databasebeta

← #139664

Subgroups — #139664

Order 512.

422 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33489C211yesmaximal, characteristic, complemented
2#28140C211yesmaximal, characteristic
3#11657C212yesmaximal, complemented
4#8616C211yesmaximal, characteristic, complemented
5#7217C212yesmaximal, complemented
6C2×C4×D8C2211yescharacteristic
7C2×C8×D4C2211yescharacteristic
8#2537—21no—
9#2169—24no—
10#2162C2212yes—
11#2112—22no—
12#2075C412yes—
13#1999C412yes—
14#1893—22no—
15#1834C2211yescharacteristic
16#1802—21no—
17#1800—21no—
18#1567C2212yes—
19#1331—24no—
20#1292—21no—
21C22×D8C2×C411yescharacteristic
22C23×C8D412yes—
23C2×#121—22no—
24C2×#121—41no—
25C2×C4×D4—21no—
26C2×C4×D4D411yescharacteristic
27C2×M32—24no—
28C4×D8—22no—
29C4×D8—22no—
30C8×D4—22no—
31C8×D4—22no—
32C2×#107C2×C411yescharacteristic
33#423—22no—
34C2×C4⋊C8D411yescharacteristic
35C2×#102—21no—
36C2×#102—22no—
37C2×#100—41no—
38C2×C22⋊C8C2×C412yes—
39C2×C4×C8C2311yescharacteristic
40#394—42no—
41#384—21no—
42#380—22no—
43#380—41no—
44C4⋊C16—22no—
45C22⋊C16—24no—
46C22⋊C16—22no—
47C4×C16D414yescomplemented
48#343—41no—
49#340—22no—
50#340—21no—
51#340C2×C412yes—
52#335—22no—
53#335—22no—
54C22×D4—21no—
55C22×D4—42no—
56C22×D4D811yescharacteristic
57C23×C4—41no—
58C23×C4C22⋊C412yes—
59C2×D8—22no—
60C2×D8—42no—
61C2×M16—42no—
62C22×C8—22no—
63C22×C8—22no—
64C22×C8—82no—
65C22×C8C22⋊C412yes—
66C22×C8C22×C411yescharacteristic
67C22×C8C2×D411yescharacteristic
68#121—44no—
69#121—82no—
70#121—84no—
71C4×D4—22no—
72C4×D4—22no—
73C4×D4—42no—
74C4×D4—42no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4D811yescharacteristic
80C2×C22⋊C4—42no—
81C2×C22⋊C4—42no—
82C2×C22⋊C4—41no—
83C2×C22⋊C4SD1612yes—
84C2×C42C2×D411yescharacteristic
85M32—84no—
86C2×C16—24no—
87#107—22no—
88C4⋊C8—22no—
89C4⋊C8—42no—
90C4⋊C8—42no—
91#102—41no—
92#102—41no—
93#102—82no—
94#100—82no—
95C22⋊C8—24no—
96C4×C8C2×D412yes—
97C4×C8#4112yes—
98#95—42no—
99C24—41no—
100C24#10212yes—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—21no—
104C2×D4—22no—
105C2×D4—21no—
106C2×D4—41no—
107C2×D4—41no—
108C2×D4—41no—
109C2×D4—41no—
110C2×D4—41no—
111C2×D4—84no—
112C2×D4—84no—
113C2×D4—162no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—41no—
119C22×C4—42no—
120C22×C4—82no—
121C22×C4—82no—
122C22×C4—81no—
123C22×C4—81no—
124C22×C4#10212yes—
125C22×C4C2×C22⋊C411yescharacteristic
126C22×C4C2×D811yescharacteristic
127C22×C4C2×SD1611yescharacteristic
128D8—24no—
129D8—42no—
130M16—82no—
131M16—82no—
132C2×C8—22no—
133C2×C8—41no—
134C2×C8—41no—
135C2×C8—44no—
136C2×C8—42no—
137C2×C8—82no—
138C2×C8—82no—
139C2×C8C4×D412yes—
140C2×C8#12011yescharacteristic
141C2×C8#12311yescharacteristic
142C4⋊C4—22no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—41no—
146C4⋊C4—41no—
147C4⋊C4—82no—
148C4⋊C4—161no—
149C22⋊C4—24no—
150C22⋊C4—84no—
151C22⋊C4—84no—
152C22⋊C4—82no—
153C42#12112yes—
154C42#13512yes—
155C16—44no—
156C23—22no—
157C23—22no—
158C23—22no—
159C23—42no—
160C23—42no—
161C23—42no—
162C23—81no—
163C23—81no—
164C23—81no—
165C23—81no—
166C23—81no—
167C23—81no—
168C23C2×#10211yescharacteristic
169D4—42no—
170D4—42no—
171D4—42no—
172D4—42no—
173D4—81no—
174D4—81no—
175D4—168no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—41no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—82no—
184C2×C4—82no—
185C2×C4—82no—
186C2×C4—81no—
187C2×C4—81no—
188C2×C4—82no—
189C2×C4—81no—
190C2×C4—162no—
191C2×C4—162no—
192C2×C4#38611yescharacteristic
193C2×C4#41611yescharacteristic
194C2×C4#46512yes—
195C2×C4#46612yes—
196C2×C4#50911yescharacteristic
197C2×C4#50912yes—
198C2×C4#51011yescharacteristic
199C8—22no—
200C8—21no—
201C8—21no—
202C8—42no—
203C8—162no—
204C22—21no—
205C22—21no—
206C22—44no—
207C22—42no—
208C22—42no—
209C22—42no—
210C22—82no—
211C22—82no—
212C22—82no—
213C22—82no—
214C22—161no—
215C22—161no—
216C22—161no—
217C22—161no—
218C22#189311yescharacteristic
219C22#213911yescharacteristic
220C22#214011yescharacteristic
221C4—21no—
222C4—22no—
223C4—42no—
224C4—162no—
225C4—161no—
226C4#189111yescharacteristic
227C4#191111yescharacteristic
228C4#221812yes—
229C4#222012yes—
230C2—21no—
231C2—21no—
232C2—82no—
233C2—162no—
234C2#948511yescharacteristic
235C2#1306111yescharacteristic
236C2#1307611yescharacteristic
subgroupG/H
center, socleC22#1893
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radical#28140C2
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #33489C2semidirect
3. #11657C2semidirect
4. #8616C2semidirect
5. #7217C2semidirect
47. C4×C16D4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#334892
#281402
#116572
#86162
#72172